Abstract
This study reports the second research cycle on students’ understanding of directional derivatives of two-variable functions. We applied Action-Process-Object-Schema (APOS) Theory as theoretical framework as in the first cycle. As a result of the first research cycle, a refined genetic decomposition describing the constructions that students may do to understand directional derivatives was proposed. Following the first cycle, we used the genetic decomposition to design didactical materials (a set of activities) to help students do the proposed constructions. The APOS theory pedagogical strategy, the ACE cycle, consisting of work on the activities in collaborative groups of three or four students, followed by whole class discussion and take-home exercises, was implemented in the classroom. In this study, we empirically test the genetic decomposition and the activity set resulting from the first research cycle by performing semi-structured interviews with a group of eleven students to see whether the GD as a model and the activity set were effective, needed to be refined, or to be rejected. We validated the genetic decomposition by showing that the didactical materials and the pedagogical method were productive and supported students’ construction of directional derivatives. Some aspects in which students’ understanding can be further improved are discussed.
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Data Availability
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
Notes
Questions I-6a and I-6b were not about directional derivatives. I-6a) What can you say about the change in the value of the function [Fig. 4] if \(x\) increases 0.02 units and \(y\) decreases 0.02 units? I-6b) Find the differential of \(f\) [Fig. 4] at the point \((1, 2)\), \(df(\mathrm{1,2})\). If it is not possible, explain why.
By the construction of “deep understanding” we mean the construction of Process or Object conceptions.
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Borji, V., Martínez-Planell, R. & Trigueros, M. University Students’ Understanding of Directional Derivative: An APOS Analysis. Int. J. Res. Undergrad. Math. Ed. (2023). https://doi.org/10.1007/s40753-023-00225-z
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DOI: https://doi.org/10.1007/s40753-023-00225-z